Производная sin(x)/(1+tan(x))

Вы ввели

  sin(x)  
----------
1 + tan(x)

$$\frac{\sin{\left (x \right )}}{\tan{\left (x \right )} + 1}$$

Подробное решение

1. Применим правило производной частного:

   $\frac{d}{d x}\left(\frac{f{\left (x \right )}}{g{\left (x \right )}}\right) = \frac{1}{g^{2}{\left (x \right )}} \left(- f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}\right)$

   $f{\left (x \right )} = \sin{\left (x \right )}$ и $g{\left (x \right )} = \tan{\left (x \right )} + 1$.

   Чтобы найти $\frac{d}{d x} f{\left (x \right )}$:

   1. Производная синуса есть косинус:

      $\frac{d}{d x} \sin{\left (x \right )} = \cos{\left (x \right )}$

   Чтобы найти $\frac{d}{d x} g{\left (x \right )}$:

   1. дифференцируем $\tan{\left (x \right )} + 1$ почленно:

      1. Производная постоянной $1$ равна нулю.

      2. Есть несколько способов вычислить эту производную.

         Один из способов:

         1. $\frac{d}{d x} \tan{\left (x \right )} = \frac{1}{\cos^{2}{\left (x \right )}}$

      В результате: $\frac{1}{\cos^{2}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right)$

   Теперь применим правило производной деления:

   $\frac{1}{\left(\tan{\left (x \right )} + 1\right)^{2}} \left(- \frac{\sin{\left (x \right )}}{\cos^{2}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right) + \left(\tan{\left (x \right )} + 1\right) \cos{\left (x \right )}\right)$

2. Теперь упростим:

   $\frac{- \sin^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}{\left(\tan{\left (x \right )} + 1\right)^{2} \cos^{2}{\left (x \right )}}$

---

Ответ:

$\frac{- \sin^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}{\left(\tan{\left (x \right )} + 1\right)^{2} \cos^{2}{\left (x \right )}}$

Первая производная

             /        2   \       
  cos(x)     \-1 - tan (x)/*sin(x)
---------- + ---------------------
1 + tan(x)                   2    
                 (1 + tan(x))     

$$\frac{\cos{\left (x \right )}}{\tan{\left (x \right )} + 1} + \frac{\left(- \tan^{2}{\left (x \right )} - 1\right) \sin{\left (x \right )}}{\left(\tan{\left (x \right )} + 1\right)^{2}}$$

Вторая производная

                                                  2                                       
            /       2   \            /       2   \             /       2   \              
          2*\1 + tan (x)/*cos(x)   2*\1 + tan (x)/ *sin(x)   2*\1 + tan (x)/*sin(x)*tan(x)
-sin(x) - ---------------------- + ----------------------- - -----------------------------
                1 + tan(x)                          2                  1 + tan(x)         
                                        (1 + tan(x))                                      
------------------------------------------------------------------------------------------
                                        1 + tan(x)                                        

$$\frac{1}{\tan{\left (x \right )} + 1} \left(- \sin{\left (x \right )} - \frac{2 \sin{\left (x \right )} \tan{\left (x \right )}}{\tan{\left (x \right )} + 1} \left(\tan^{2}{\left (x \right )} + 1\right) - \frac{2 \left(\tan^{2}{\left (x \right )} + 1\right) \cos{\left (x \right )}}{\tan{\left (x \right )} + 1} + \frac{2 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \sin{\left (x \right )}}{\left(\tan{\left (x \right )} + 1\right)^{2}}\right)$$

Третья производная

                         3                         2                                                  2                                                                                           2              
            /       2   \             /       2   \             /       2   \            /       2   \             /       2   \                      2    /       2   \             /       2   \               
          6*\1 + tan (x)/ *sin(x)   2*\1 + tan (x)/ *sin(x)   3*\1 + tan (x)/*sin(x)   6*\1 + tan (x)/ *cos(x)   6*\1 + tan (x)/*cos(x)*tan(x)   4*tan (x)*\1 + tan (x)/*sin(x)   12*\1 + tan (x)/ *sin(x)*tan(x)
-cos(x) - ----------------------- - ----------------------- + ---------------------- + ----------------------- - ----------------------------- - ------------------------------ + -------------------------------
                           3               1 + tan(x)               1 + tan(x)                          2                  1 + tan(x)                      1 + tan(x)                                  2         
               (1 + tan(x))                                                                 (1 + tan(x))                                                                                   (1 + tan(x))          
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                    1 + tan(x)                                                                                                   

$$\frac{1}{\tan{\left (x \right )} + 1} \left(- \cos{\left (x \right )} - \frac{2 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \sin{\left (x \right )}}{\tan{\left (x \right )} + 1} - \frac{4 \sin{\left (x \right )} \tan^{2}{\left (x \right )}}{\tan{\left (x \right )} + 1} \left(\tan^{2}{\left (x \right )} + 1\right) + \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right) \sin{\left (x \right )}}{\tan{\left (x \right )} + 1} - \frac{6 \cos{\left (x \right )} \tan{\left (x \right )}}{\tan{\left (x \right )} + 1} \left(\tan^{2}{\left (x \right )} + 1\right) + \frac{12 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \sin{\left (x \right )}}{\left(\tan{\left (x \right )} + 1\right)^{2}} \tan{\left (x \right )} + \frac{6 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \cos{\left (x \right )}}{\left(\tan{\left (x \right )} + 1\right)^{2}} - \frac{6 \left(\tan^{2}{\left (x \right )} + 1\right)^{3} \sin{\left (x \right )}}{\left(\tan{\left (x \right )} + 1\right)^{3}}\right)$$